Problem 65 - Algebra A Combined Function

Question

Simplify each expression. Then determine whether the given answer is correct. $$ \frac{9-x^{2}}{x-3} ; \text { Answer: }-3-x $$

Step-by-Step Solution

Answer

The given answer \(-3-x\) is correct.

Step 1: Factor the Numerator

The numerator is \(9 - x^2\). Recognize this as a difference of squares: \(9 - x^2 = (3 - x)(3 + x)\). This is a special factoring pattern where \(a^2 - b^2 = (a-b)(a+b)\).

Step 2: Set Up the Simplified Expression

Now that the numerator \(9 - x^2\) is factored into \((3-x)(3+x)\), the expression becomes \(\frac{(3-x)(3+x)}{x-3}\).

Step 3: Cancel Common Factors

Rewrite the expression to see if cancellation of terms is possible: \((3-x) = -(x-3)\). Therefore, the expression \(\frac{(3-x)(3+x)}{x-3}\) becomes \(-\frac{(x-3)(3+x)}{x-3}\). The \(x-3\) terms cancel out, resulting in \(-(3+x)\).

Step 4: Simplify the Result

\(-(3+x)\) can be rewritten by distributing the negative sign: \(-3 - x\).

Step 5: Verify Given Answer

The simplified form of the expression is \(-3 - x\), which matches the given answer \(-3-x\). Hence, the given answer is correct.

Key Concepts

Difference of Squares

The concept of the difference of squares is a key algebraic pattern that helps in simplifying expressions. It occurs when you have two squared terms separated by a minus sign, such as \(a^2 - b^2\). This pattern can be factored into the form \((a-b)(a+b)\).
For example, the expression \(9 - x^2\) is a difference of squares because it can be written as \((3)^2 - (x)^2\). By recognizing this pattern, you can quickly factor \(9 - x^2\) into \((3-x)(3+x)\).
This tool is extremely useful because it simplifies complex algebraic expressions, making them easier to handle in further operations. Always look for this pattern when you see a subtraction between two squares; it is a shortcut that can save you time.

Factoring

Factoring is the process of breaking down an expression into simpler multipliers that can be multiplied together to give the original expression.
In our example, we factor the numerator \(9-x^2\) into \((3-x)(3+x)\) by recognizing it as a difference of squares.
When factoring, ask yourself: "Can this expression be broken down into simpler parts?" If you identify patterns like difference of squares or other factoring techniques, apply them. Factoring allows you to simplify expressions and make calculations easier.

Cancellation of Terms

Cancellation of terms is a crucial step that can further simplify a fraction. After factoring, review your expression to see if there are any common terms in the numerator and the denominator.

• In the expression \(\frac{(3-x)(3+x)}{x-3}\), note that \(3-x\) is equivalent to \(-(x-3)\).
• This means we can rewrite the fraction as \(-\frac{(x-3)(3+x)}{x-3}\).
• Because \(x-3\) appears in both the numerator and denominator, it can be cancelled out, leaving behind \(-(3+x)\).

After cancelling, simplify further by distributing any remaining factors, like the negative sign, to get the final result. This step helps consolidate the expression into its simplest form, making it easier to interpret or use in further calculations.